Artículos de revistas
The V-density Of Eigenvalues Of Non Symmetric Random Matrices And Rigorous Proof Of The Strong Circular Law
Registro en:
Random Operators And Stochastic Equations. , v. 5, n. 4, p. 371 - 406, 1997.
9266364
2-s2.0-33845598984
Autor
Girko V.L.
Institución
Resumen
We review some results obtained in series of papers on non Hermitian random matrices in some problems of spin glasses and neural nets. We present new theory of such matrices on the basis of the V-transform of normalized spectral function (n.s.f.) ν n(x,y) of the eigenvalues of non symmetric matrix Ξ with n.s.f. μ n(x,τ) of the eigenvalues of the Hermitian G-matrix (Ξ-τI) (Ξ - τI) *, τ = t + is: (Formula Presented) where ε > 0. This article discusses methodological approach which allows one to obtain rigorous proof of the strong Circular law and to describe the region where the eigenvalues of large non Hermitian random matrices are distributed. © 1997 VSP. 5 4 371 406 Bronk, B.V., Accuracy of the semicircle approximation for the density of eigenvalues of random matrices (1964) J. Math. Phys., 5, pp. 215-220 Bronk, B.V., Exponential ensemble for random matrices (1965) Math. Phys., 6, pp. 228-237 Porter, C.E., (1965) Statistical Theories of Spectra: Fluctuations, Collection of Re Prints and Original Papers, , Academic Press, New York Dyson, F.I., The dynamics of disordered linear chain (1953) Phys. Rev., 92 (6), pp. 1331-1338 Dyson, F.L., A Brownian-motion model for the eigenvalue of a random matrix (1962) Math. Phys., 3 (6), pp. 31-60 Dyson, F.J., Correlations between eigenvalues of a random matrix (1970) Commun. Math. Phys., 19, pp. 235-250 Ginibre, J., Statistical ensembles of complex, quaternion, and real matrices (1965) J. Math. Phys., 6, pp. 440-449 Mehta, M.L., (1967) Random Matrices and the Statistical Theory of Energy Levels, , Academic Press, New York, London Evangelou, S.N., Katsanos, D.E., Energy level statistics in disordered metals with an anderson transition (1996) Jornal of Statistical Physics, 85 (5-6), pp. 525-550 Forrester, P.J., Zuk, J.A., Applications of the Dotsenko-Fateev integral in random-matrix models (1966) Nuclear Physics B, 473, pp. 616-630 Fox, D., Kahn, P.B., Identity of the n-th order spacing distributions for a class of Hamiltonian unitary ensembles (1964) Phys, B.ev., 134, pp. B1151-B1192 Schmidt, H., Disordered One-dimensional Crystals (1957) Phys. Rev., 105 (2), pp. 425-441 Lifshits, I.M., On the structure of the energy spectrum and quantum state of disordered condenser systems (1965) Sov. Fiz. Usp., 7, pp. 549-573. , in Russian Tracy, A.C., Widom, H., Level spacing distributions and the Airy kernel (1994) Commun. Math. Phys., 159, pp. 151-174 Tracy, A.C., Widom, H., Level spacing distributions and the Bessel kernel (1994) 70mmun. Math. Phys., 161, pp. 289-309 Tracy, A.C., Widom, H., Fredholm determinants, differential equations and matrix models (1994) Commun. Math. Phys., 163, pp. 33-72 Tracy, A.C., Widom, H., On orthogonal and symplectic matrix ensembles (1996) Commun. Math. Phys., 177, pp. 727-754 Wigner, E.P., Statistical properties of real symmetric matrices with many dimensions (1957) Can. Math. Proc., pp. 174-198 Wigner, E.P., On the distribution of the roots of certain symmetric matrices (1958) Ann. Math., 67 (2), pp. 325-327 Wigner, E.P., Random matrices in physics (1967) SIAM Rev., 9 (1), pp. 1-23 Zakrewski, I., Dupret, K., Delande, D., Wigner or Non-Wigner: That Is the Question (1995) Proceedings of the XXXIst Winter School of Theoretical Physics "chaos-the Interplay between Stochastic and Deterministic Behavior", pp. 559-564. , Springer [Gir35] spectral theory of random matrices (1985) Russian Math. Surveys, 40 (1), pp. 77-120 Kuz, M., Lewentein, M., Haake, F., Density of eigenvalues of random band matrices (1991) Phys. Rev. A, 44 (5), pp. 2800-2808 Girko, V.L., Preston, N., Numerical and Monte Carlo Verification of the First Spacing Law (1996) Random Operators and Stochastic Equations, 4 (4), pp. 303-314 Arnold, L., On the asymptotic distribution of the eigenvalues of random matrices (1960) Math. Anal. Appl., 20, pp. 262-268 Arnold, L., On wigners semicircle law for the eigenvalues of random matrices (1971) Z. Wahr. Verw. Geb., 34, pp. 191-198 Berezin, F.A., Some remarks on Wigner distribution (1973) Theoret. Math. Phys., 17 (3), pp. 1163-1175 De Monvel, B., Pastur, L., Shcerbina, M., On the statistical mechanics approach in the random matrix theory: Integrated density of states (1995) Stat. Phys., 79 (3-4), pp. 585-613 De Monvel, B., Khorunzhy, A., Vasilchuk, V., Limiting eigenvalue distribution of random matrices with correlated entries (1996) Markov Processes Relat. Fields, 2, pp. 607-636 Brody, T.A., Flores, J., French, J., Mello, P.A., Pandey, A., Wong, S., Randommatrix physics: Spectrum and strength fluctuation (1981) Rev. Modern Phys., 58 (3), pp. 385-479 Camarda, H.S., Georgopulos, P.P., Statistical behaviour of atomic energy levels: Aqreement with random matrix theory (1983) Phys. Rev. Letters, 50, pp. 492-495 Casati, G., Molinari, G.L., Izrailev, F.M., Scaling properties of band random matrices (1980) Phys. Rev. Lett., 64, pp. 1851-1854 Constantinescu, F., Felder, G.M., Gawedzki, K., Kupiainen, A., Analyticity of density of states (1987) Stat. Phys., 48 (3), pp. 365-391 Girko, V.L., Casati, G., Molinari, L., Equation for limit spectral functions of band random matrices (1993) Random Oper. Stock. Eqs., 1 (1), pp. 1-6 Pastur, L.A., The Spectra of Random Selfadjoint Operators (1973) Russian Math. Surveys, 28 (1), pp. 1-67 Girko, V.L., Casati, G., Generalized Wigner law for band random matrices (1993) Random Oper. Stock. Eqs., 1, pp. l Girko, V.L., Casati, G., Limit theorems for band random matrices whose entries have bounded variances (1993) Random Oper. Stock. Eqs., 1 (2), pp. 181-191 Girko, V.L., Kirsch, W., Kutzelnigg, A., Necessary and sufficient conditions for the semicircle law (1994) Random Oper. Stock. Eqs., 2 (2), pp. 195-202 Opperman, R., Wegner, F., Disordered System with N Orbitals per Site: N - I Expansion (1979) Z. Phys., 34, pp. 327-348 Khorunzhy, A., Pastur, L., Limits of infinite interaction radius, dimensionality and the number of components for random operators with off-diagonal randomness (1993) Commun. Math. Phys., 153, pp. 605-646 Khorunzhy, A., Khoruzhenko, B., Pastur, L., On the ln corrections to the green functions of random matrices with independent entries. of physics a (1995) Mathematical and General., 28, pp. L31-L35 Khorunzhy, A., Khoruzhenko, B., Pastur, L., Asymptotic properties of large random matrices with independent entries (1996) J. Math. Phys., 37 (10), pp. 5033-5060 Khorunzhy, A., Molchanov, S., Pastur, L., On the eigenvalue distribution of band random matrices in the limit of their infinite order (1992) Teor. Mat. Fiz., 90, pp. 163-178 Marchenko, V.A., Pastur, L.A., Distribution of the eigenvalues in certain sets of random matrices (1967) Mat. Sb., 1, pp. 457-483. , in Russian Pastur, L.A., On the universality of the level spacing distribution for some ensemble of random matrices (1992) Lett. Math. Phys., 25, pp. 259-265 Pastur, L.A., Shcherbina, M., Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles (1997) Stat. Phys., 86, pp. 109-147 Pastur, L.A., Figotin, A., (1992) Spectra of Random and Almost Periodic Operators, , Springer Verlag Schafer, L., Wegner, F., Disordered System with N Orbitals per Site: Lagrange Formulation (1980) Z. Phys., 38, pp. 113-126 Sommers, H.J., Crisanti, A., Sompolinsky, H., Stein, Y., Spectrum of large random asymmetric matrices (1988) Phys. Rev. Lett., 60, pp. 1895-1898 Khorunzenko, B., Large-TV eigenvalue distribution of randomly perturbed asymetric matrices (1996) Phys. A: Math. Gen., 29, pp. L165-L169 Fyodorov, Y.V., Khoruzenko, H.J.B., Sommers, Almost-Hermitian random matrices: Eigenvalue density in the complex plane (1997) Physics Letters A 226, 226 (1-2), pp. 46-51 Haake, F., (1991) Quantum Signatures of Chaos, , Springer Verlag, Heidelberg Izrailev, F.M., Simple models of quantum chaos: Spectrum and eigenfunctions (1990) Phus. Report., 196 (5-6), pp. 299-392 Guhr, T., Transitions toward quantum chaos: With supersymmetry from Poisson to Gauss (1996) Annals of Physics, 250, pp. 145-192 Bessis, D., Itzykson, C., Zuber, J., Quantum field theory techniques in graphical enumeration (1980) Adv. Appl. Math., 1, pp. 109-157 Brezin, E., Itzykson, C., Paris, G., Zuber, J., Planar diagrams (1978) Commun. Math. Phys., 59, pp. 35-51 Brezin, E., Zee, A., Universality of the correlations between eigenvalues of large random matrices (1993) Nucl Phys. B., 402, pp. 613-627 Demetrefi, K., Two-dimensional quantum gravity, matrix models and string theory (1993) Int. J. Mod. Phys. A., 8, pp. 1185-1244 Franchesco, P.D., Ginsparg, R., Zinn-Justin, I., 2D gravity and random matrices (1995) Phys. Rep., 254 (1) Fyodorov, Y.V., Alexander, D.M., Scaling properties of localization in random band matrices: A s-model approach (1991) Phys. Rev. Lett., 67, pp. 2405-2409 Fyodorov, Y.V., Sommers, H.J., Statistics of resonance poles, phase shifts ant time delays in quantum chaotic scattering: Random matrix approach for systems with broken time-reversal invariance (1997) Journal of Mathematical Physics, 38 (4), pp. 1918-1981 Bogomolny, E., Bohigas, O., Leboeuf, P., Distribution of roots of random polynomials (1992) Phys. Rev. Lett, 68, pp. 2726-2729 Bogomolny, E., Bohigas, O., Leboeuf, P., Quantum chaotic dynamics and random polynomials (1996) Journal of Statistical Physics, 85 (5-6), pp. 639-679 Fernandez, R., Frölich, J., Sokal, A., (1992) Random Walks Critical Phenomena and Triviality in the Quantum Field Theory, , Springer Verlag, Heidelberg Girko, V.L., Ezhov, S.N., Pluyko, V.A., Integral equations for average scattering matrix (1988) Ukr. Fiz. Zh., 33 (10), p. 1451. , in R.ussian Girko, V.L., Ezhov, S.N., Pluyko, V.A., The integral equations for the average stochastic scattering matrix (1993) Random Oper. Stock. Eqs., 1 (2), pp. 161-170 Girko, V.L., Olhovsky, V.S., Chinarov, V.A., To the theory of fluctuation of unitary 5-matrix for nuclear reactions (1984) Izv. Akad. Nauk SSSR, 48 (1), pp. 166-171. , in Russian Aizenman, M., Lebowitz, L.L., Ruelle, D., Some Rigorous Results on the Shering ton-Kirkpatrick Spin Glass Model (1987) Commun. Math. Phys., 112, pp. 3-20 Hertz, L.A., Statistical dynamics of learning, in: Lecture Notes in Physics (1990) Statistical Mechanics of Neural Networks, Proceedings, Sitges, , Ed. Luis Garrido Barcelona, Spain, Springer-Velag Crisanti, A., Sompolinsky, H., Dynamics of spin systems with randomly asymmetric bonds: Langevin dynamics and a spherical model (1987) Phys. Rev. A., 36 (10), pp. 4922-4939 Lehmann, N., Sommers, H.J., Eigenvalue statistics of random real matrices (1991) Phys. Rev. Lett., 67 (8), pp. 941-944 Mezinescu, G.A., Bessis, D., Fournier, J.D., Manticaand, G., Aaron, F.D., Distribution of Roots of Random R.eal Generalized Polynomials (1997) Journal of Statistical Physics, 86 (3-4), pp. 59-71 Girko, V.L., (1975) Random Matrices, , Kiev, Published by Kiev University in Russian Girko, V.L., V-Transform (1982) Reports AN USSR, Series of Mathematics and Mechanics, 3, pp. 5-6 Girko, V.L., On the circle law (1983) Theory of Probability, Mathematical Statistics, 28, pp. 15-23 Girko, V.L., The Circle Law (1984) Theory of Probabilities and Their Applications, 29 (4), pp. 669-679 Girko, V.L., Regularized V-transform (1984) Theory of Probability and Its Applications, 29 (2), pp. 416-417 Girko, V.L., (1985) The Elliptic Law, pp. 171-173. , Reports of the Fourth Vilnius Conference on the Theory of Probability and Mathematical Statistics, Vilnius Girko, V.L., The Elliptic Law (1985) Doclady AN USSR, Ser. Matematika i Mehanika, 1, pp. 3-7 Girko, V.L., The Elliptic Law (1985) Theory of Probability and Their Applications, 30 (4), pp. 640-651 Girko, V.L., The Elliptic Law and Elements of G-analysis (1987) Proceedings of the Fourth Vilnius Conference, 1, pp. 489-509. , VNU Science Press BV Girko, V.L., (1990) Theory of Random Determinants, , Kluwer Academic Publishers Girko, V.L., Canonical spectral equation for singular spectral functions of random matrices (1993) Russian Math. Surveys., 48 (3), pp. 163-180 Girko, V.L., The Circular Law: Ten Years Later (1994) Random Oper, and Stock. Equ., 2 (3), pp. 235-277 Girko, V.L., The Elliptic Law: Ten Years Later i (1995) Random Operators and Stochastic Equations, 3 (3), pp. 257-302 Girko, V.L., The Elliptic Law: Ten Years Later II (1995) Random Operators and Stochastic Equations, 3 (4), pp. 377-398 Girko, V.L., (1996) Theory of Linear Algebraic Equations with Random Coefficients, p. 320. , Allerton Press, Inc. New York Girko, V.L., Strong Circular Law (1997) Random Operators and Stochastic Equations, 5 (2), pp. 173-196 Girko, V.L., Strong Elliptic Law (1997) Random Operators and Stochastic Equations, 5 (3), pp. 269-302 Bai, Z.D., Circular Law (1997) Ann. Probab., 25 (1), pp. 494-529 Von Bahr, L., Esseen, C.G., Inequalities for the r-th absolute moment of a sum of ranom variables, 1 < r < 2 (1965) Ann. Math. Statist., 36 (1), pp. 299-303 Burkholder, D.L., Distribution function inequalities for martingales (1973) Ann. Probab., 1 (1), pp. 19-42 Muskhelishvili, N.I., (1953) Singular Integral Equations, , Noordhoff, Groningen Muskhelishvili, N.I., (1966) Some Principal Problems of Mathematical Theory of Elasticity, , Nauka, Moscow in Russian Rodgers, G.J., Bray, A.J., Density of states of a sparse random matrices (1988) Phys. Rev. B, 37, pp. 3557-3762 Rodgers, G.J., Dominicis, C., Density of states of a sparse random matrices (1990) Phys. A, 23, pp. 1567-1573 Khorunzhy, A., Rodgers, G.J., Eigenvalue distribution of large dilute random " matrices (1997) J. Math. Phys., 38 (6), pp. 3300-3320 Sompolinsky, H., Neural networks with non-linear synapces and static noise (1986) Phys, Rev. A, 34, pp. 2571-2574 Moro, L., Burke, J.V., Overton, M.L., On the Lidskii-Vishik-Lyusternik Perturbation Theory for Eigenvalues of Matrices with Arbitrary Jordan Structure (1997) SI AM Journal on Matrix Analysis and Applications, 18 (4) Guhr, T., Wettig, T., An itzykson-zuber-like integral and diffusion for complex ordinary and supermatrices (1996) J. Math. Phys., 37 (12), pp. 6395-6413 Itzykson, C., Zuber, J.B., The planar approximation II (1980) Math. Phys., 21 (3), pp. 411-421